2 '( ) / ( )RT gn x n xγ =For 1:ik ρ =

When a heavy fluid is supported by a light fluid, the system is RT unstable.

Rayleigh-Taylor (RT) Instability

FLR Stabilize RT with Large ikρAccording to Roberts and Taylor[1], the dispersion relation of RT changes as finite larmor radius (FLR) effect is considered:

Figure. 1

Velocity Shear Stabilization of RT[2,3]

It is also known that velocity shear can stabilize RT-instability by reducing the length scale (increase k) of the fluctuations of the system, so that viscosity effect becomes important.

ˆ'ov V xy=v

The Question

Since k can be increased by velocity shear, could FLR effect be more effective in stabilizing RT-instability by velocity shear?

System of Equations[1]

218t

Bnd u nT ng

Mν

π

= − ∇ + + + ∇ ⋅ Π

v v

( ) 0tn nu∂ + ∇ ⋅ =v

0u∇ ⋅ =v

(1)

(2)

(3)where is the gyroviscosity parameter representing the FLR effect and

( ) ( ) ( )( ) ( ) ( )

,

.

x x y y x y x x y yx

y x y y x x x x y yy

n u u n u u

n u u n u u

∇ ⋅ Π = ∂ ∂ + ∂ − ∂ ∂ − ∂ ∇ ⋅ Π = −∂ ∂ + ∂ − ∂ ∂ − ∂

2 / 4i iν ρ= Ω

Also B is in the z-direction only and system is symmetric in the z-direction.

Perturbation of Static Equilibrium

( ) ,

0 .

xo o

o

n x n e

v

κ==v

According to Roberts and Taylor, the equilibrium is assumed to be:

Taking the curl of (2) to eliminate B, linearizing and assuming solutions of the form and weak x-dependence, the dispersion relation is found to be (see Figure 1.):

( )exp i t ikyω +

2*( )i RTω ω ω γ+ = −

It is clear that RT is stabilized for large .* 2i kω νκ≡

Perturbation of Equilibrium with Linear Velocity Shear

We consider now a linear velocity profile in equilibrium:

0 ˆ'v V xy=v'Vwhere is a constant. After taking the curl of (2) and linearizing

the equations, a transformation of coordinates is done to take into account the V-shear:

Assume weak -dependence …

… and fourier transform .

ζ ⊥

ζ

'y V xt

ζ ττ

= −=

Time Evolution Equation without FLR[3]

Before going further for the case of FLR-assisted V-shear calculation, as a comparison for later discussion, we would like to present the result for the case of V-shear without FLR but viscosity:

( )( )

2

2 2

0 ,

,

t

t y

d D n

n d g nµ φ

⊥

⊥ ⊥

− ∇ =

− ∇ ∇ = ∂

ˆwhere 't t yd V x z φ≡ ∂ + ∂ + ×∇ ⋅∇

: density: perturbed electrostatic potential:diffusion coefficient: kinmatic viscosity

n

Dφ

µ

2 2ˆ ˆ

ˆ(1 )t td t d X Xα + =

( ) ( )2

ˆ( ) 2 20

ˆ ˆ ˆwhere exp and ( ) 1 / 3h t

RT

kn n x Xe h t t t

µκ α

γ−= + ≡ +

(4)

'ˆ , , , ,RTRT RT

V kt

kνκ κ

γ τ α β σγ γ

≡ ≡ ≡ ≡

and making the assumptions , we can eliminate all the -related terms and obtain a two-parameter time evolution equation for :

1, 1and 1α β σα≥ = =σ

( )( ) ( )

2 2ˆ ˆ

2 2 2ˆ

ˆ ˆ1 ( )

ˆ ˆ ˆ2 1 ( ) 1 2 0

t t

t

d t d n t

i t d n t i t n

α

β α βα

+ − + − + =

%% %

(*)

n%

Time Evolution Equation with FLRReturning to FLR-assisted V-shear case, by introducing several normalized parameters as follows,

It is worth to note the difference and similarity between (4) and (*) .

Asymptotic Solution• Solve (*) in short- and long-time scales separately• Match the corresponding solutions.Short-time Behavior

2

1ˆ ,1tβα

=

( )2 2ˆ ˆ

ˆ1 0t td t d n nα < < + − = % %

which admits the solution:

ˆ ˆ( ) ( )w wn A P i t BQ i tα α< = +%t

~ wtn<%

1/ β0

2

1 1 4where and are Legendre Polynomials of degree and 1 .

2 2w wP Q w wα

= − + +

For , (*) can be written as

Long-time Behavior

For , (*) can be written asˆ ˆ1and ~1t tα β>>

( ) ( ).. .

2 2 2 2 2 2ˆ ˆ ˆ ˆ2 1 2 0t n t i t n i t nα α βα βα> > >+ − − + =% % %and it’s general solution is given by

ˆ ˆ

1/2 1 /2ˆ ˆ( ) ( )

ˆ ˆ

i t i t

w w

e en C J t D Y t

t t

β β

β β> + += +%

where Jm and Ym are bessel functions of integer orderand w is defined as before. 1/2~ t−

~ if 0wt D =

t

n>%

1/α0

Asymptotic Matching

10.20.01(only for numerical solution)

αβσ

===

t

0

nn%%

• Initial Conditions è Determine A and B• Matching and è D = 0, relate A and C

An approximate global solution is obtained

n< n>

0

nn%%

t

… the peak value might grow to a too large value that linear calculation cannot be applied.

The perturbation dies out in the long run but …

Therefore we declare as a condition for stabilization.

Using some identities of the Bessel functions, the criterion is found to be:

1 /2ln'

ln 2RT

RV ν

ν γ ≥

3 .RTR

kν

γνκ

=, where

Stabilization Criterion

(5)

max

0

2nn

<%%

Stabilization Criteria Comparison

where . Since , it is foundthat FLR (i.e. Eq.(5)) would require a less demanding V’ to stabilize the system.

From the solution of Equation (4), a stabilization criterion can also be obtained for the case without FLR but kinematic viscosity. It read[3]:

' ln ,RTV Rµ µγ≥ (6)

/ ~ 1i iR Rµ ν ω τ σ >>2/RTR kµ γ µ=

6.63Fusion Parameters[6]

3.72.8MCX—Present Parameters[5]

Viscosity-assistedFLR-assistedPlasma System

Real SystemsFor system in which magnetic curvature exists, the effective gravity is taken to be , where R and a are some minor radius and majorradius respectively, so the corresponding growthrate . In unit of (i.e.~1/soundtime!!), we have[4] :

2 2,~ /th ig u a R

,~ /RT th iau Rγ κ RTγ'

criticalV

[1] K.V. Roberts and J.B. Taylor, PRL 8, 197 (1962)[2] H. Biglari, P.H. Diamond and P.W. Terry, Phys. Fluids B 2, 1 (1990)[3] A.B. Hassam, Phys. Fluids B 4, 485 (1992)[4] Criteria (6) is a good order of magnitude approximation for the viscosity-assisted

case. But a more precise calculation has been done instead of using (6) directly for a fair comparison purpose.

[5] MCX (Maryland Centrifugal Experiment)—present parameters: T ~ 30 eV, n ~ 1020 m-3, B ~ 0.1 T, R ~ 2 m, a ~ 0.3 m.

[6] Fusion parameters:T ~ 10 keV, n ~ 1020 m-3, B ~ 1 T, R ~ 3 m, a ~ 1 m.